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On Inequivalent Representations of Matroids Over Finite Fields* James Oxley and Dirk Vertigan

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On Inequivalent Representations of Matroids Over Finite Fields* James Oxley and Dirk Vertigan Journal of Combinatorial Theory, Series B TB1695 journal of combinatorial theory, Series B 67, 325343 (1996) article no. 0049 On Inequivalent Representations of Matroids over Finite Fields* James Oxley and Dirk Vertigan Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803-4918 and Geoff Whittle Department of Mathematics, Victoria University, P.O. Box 600, Wellington, New Zealand Received May 30, 1995 Kahn conjectured in 1988 that, for each prime power q, there is an integer n(q) such that no 3-connected GF(q)-representable matroid has more than n(q) inequivalent GF(q)-representations. At the time, this conjecture was known to be true for q=2 and q=3, and Kahn had just proved it for q=4. In this paper, we prove the conjecture for q=5, showing that 6 is a sharp value for n(5). Moreover, we also show that the conjecture is false for all larger values of q. 1996 Academic Press, Inc. 1. INTRODUCTION In the study of representations of matroids over finite fields, the problem of inequivalent representations arises almost immediately. For example, consider the 9-point rank-3 matroid M whose only non-trivial lines are three disjoint 3-point lines. For a large enough field F, the matroid M can be represented by a set of points in which the non-trivial lines are copunctual, and can also be represented by a set of points in which they are not; see Fig. 1. Now, automorphisms of projective planes preserve copunctuality, so it is clear that the two representations of M are inequivalent for any natural notion of equivalence of representations. For small enough fields, this problem does not arise. It is easily seen that GF(2)-representations of a matroid are equivalent, and Brylawski and Lucas [4] have shown that GF(3)-representations are unique. Kahn [9] proved that GF(4)-representations are unique for 3-connected matroids. * This paper is dedicated to Don Row who introduced all three authors to matroids. 325 0095-8956Â96 18.00 Copyright 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. File: 582B 169501 . By:CV . Date:31:07:96 . Time:15:09 LOP8M. V8.0. Page 01:01 Codes: 3973 Signs: 2041 . Length: 50 pic 3 pts, 212 mm 326 OXLEY, VERTIGAN, AND WHITTLE Figure 1 However, for q>4, one cannot guarantee uniqueness of representations, even for 3-connected matroids. Given this, one could at least hope that, for such fields, the number of inequivalent representations was limited in some way. Indeed, in [9], Kahn conjectured that, for each prime power q, there is an integer n(q) such that no 3-connected GF(q)-representable matroid has more than n(q) inequivalent representations. In this paper, we prove Kahn's conjecture for q=5 showing that there are at most six inequivalent GF(5)-representations of a 3-connected matroid. We also provide counter- examples to show that Kahn's conjecture is false for all larger values of q. Kahn's proof that GF(4)-representations of 3-connected matroids are unique uses an elegant geometric argument. A crucial result needed in this argument is the following theorem of Seymour [19]. A matroid N uses a set X if XE(N). Theorem. (1.1) If x1 and x2 are elements of a 3-connected nonbinary matroid M, then M has a U2, 4 -minor using [x1 , x2]. Our proof of Kahn's conjecture for GF(5) is structured similarly to Kahn's proof for GF(4). Section 3 is devoted to proving a result analogous to Theorem 1.1. The crucial fact needed is that if M is a 3-connected GF(5)- representable matroid with a U2, 4-restriction and a U2, 5-minor, then M has a U2, 5-minor using the elements of the U2, 4-restriction. This fact, stated as Corollary 3.10, follows immediately from more general theorems proved in Section 3. Both Theorem 1.1 and the results of Section 3 are examples of ``roundedness'' results. A brief discussion of such results and their role in matroid theory is included in Section 2. The proof of the conjecture for GF(5), given in Section 4, then applies Corollary 3.10, using File: 582B 169502 . By:SD . Date:28:06:96 . Time:07:57 LOP8M. V8.0. Page 01:01 Codes: 2290 Signs: 1775 . Length: 45 pic 0 pts, 190 mm ON INEQUIVALENT REPRESENTATIONS 327 a similar geometric argument to Kahn's proof for GF(4). Not surprisingly, some complications arise which are peculiar to the GF(5) case. The counterexamples to Kahn's conjecture for q7 are given in Section 5. We present two classes of counterexamples. The first class deals with the case where the multiplicative group of GF(5) has a proper subgroup of order at least three. The only cases not covered by the first class of counter- examples occur when q=2k for some k>2 such that 2k&1 is prime. But, in these cases, the additive group of the field has a proper subgroup of order at least three. The second class of counterexamples includes this case. It is of interest to ask if the results in this paper have any implications for a longstanding conjecture of Rota [14] that the set of forbidden minors for representations over GF(q) is finite. This conjecture has only been proved when q # [2, 3] and, recently [7], when q=4, and all known proofs in these cases make essential use of unique representability [1, 8, 10, 17, 21, 22]. We do not know if there is any connection between the results of this paper and the validity of Rota's conjecture in general. However, it is clear that any proof of Rota's conjecture for a prime power q>5 would require significantly different techniques from those that have been used to prove the conjecture for q4. 2. PRELIMINARIES Familiarity is assumed with the elements of matroid theory. Throughout, the terminology follows [12]. We include here a brief discussion of those aspects of matroid theory that are of particular importance for this paper. Representations A matroid M is representable over a field F or F-representable if there is a function . from the ground set of M into a vector space V over F which preserves rank. In other words, rM(X )=rV[.(x): x # X] for all XE(M). Equivalently, M is F-representable if and only if there is a matrix A over F whose columns are labelled by the elements of M such that, for all XE(M), the matrix consisting of the columns of A that are labelled by elements of X has rank equal to rM(X ). Such a matrix is called a represen- tation of M. It is clear that a matroid is F-representable if and only if the associated simple matroid is F-representable. Thus, when considering representability questions, one frequently restricts attention to simple matroids, and we shall do this from now on. In this case, the function . above may be viewed as a one-to-one function into a projective space over F. Indeed, it is commonplace to think of a rank-r simple F-representable matroid M as being embedded in the projective space PG(r&1, F), that is, as being a File: 582B 169503 . By:CV . Date:31:07:96 . Time:15:09 LOP8M. V8.0. Page 01:01 Codes: 3116 Signs: 2655 . Length: 45 pic 0 pts, 190 mm 328 OXLEY, VERTIGAN, AND WHITTLE restriction of PG(r&1, F). Two such embeddings .1 and .2 are equivalent if there is an automorphism % of PG(r&1, F) such that %(.1(e))=.2(e) for all e in E(M). Let |E(M)|=n. Since there is a natural way to associate such an embedding with each r_n matrix representation of M over F, this defines equivalence of two such matrix representations. This definition can be translated into purely matrix terms as follows. First, for r # [1, 2], the automorphism group of PG(r&1, F) is the symmetric group. Thus if r(M)=r2, then all r_n matrix representations of M over F are equiv- alent. If r(M)>2, it follows from a theorem, sometimes known as the Fundamental Theorem of Projective Geometry, that two r_n matrix representations are equivalent if one can be obtained from the other by a sequence of the following operations. (For details, see [12, Section 6.3].) (i) Interchange two rows. (ii) Multiply a row by a non-zero member of F. (iii) Replace a row by the sum of that row and another. (iv) Interchange two columns (moving their labels with the columns). (v) Multiply a column by a non-zero member of F. (vi) Replace each entry of the matrix by its image under some automorphism of F. We say that M is uniquely representable over F if all r_n representations of M over F are equivalent. The fact that, by the above definition, all representations of a rank-2 matroid are equivalent has the disconcerting consequence that a matroid M and its dual may have different numbers of inequivalent representations, although this can only occur if r(M)orr(M*) is 2. One could remedy this by modifying the definition so that, for representations of rank-2 matroids to be equivalent, they must be obtainable from each other via a sequence of operations (i)(vi). Such a change does not alter the results of this paper, and we prefer to follow Kahn [9] and maintain the link between equivalence and automorphisms of the underlying projective geometry.
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